Option greeks: sensitivity to 1% move

In a Black&Scholes framework how can I compute the following sensitivities:

• to 1% move in the underlying price
• to 1% move in implied volatility

I would like the greeks to tell me how many dollars I lose/gain if the underlying/implied volatility moves by 1%. In particular, I would like to calculate the delta and gamma (to 1% move in underlying price) and vega and volga (to 1% move in implied volatility).

For the vanna I would like to consider a 1% move in both underlying and implied volatility.

Can you please suggest how to modify Black&Sholes greeks and also how to compute the sensitivities numerically?

A reference would also be very welcome. Thank you.

2 Answers

Try Finite Differences to calculate your Greeks, it will give all the greeks for that specific underlying moviment. In order to back out the dollar change in your pnl just multiply each greek by the amount held in that position.

In the BS model, everything is explicit.

If your spot increases by $h\%$, the price will increase by $\Delta_{rel,h}\%$ where $$\Delta_{rel,h} = \frac{C_{BS}(S(1+h),T,K,\sigma)}{C_{BS}(S,T,K,\sigma)} - 1$$ That's high school math.

• Thank you. Can you suggest a closed-form formula also? – mickG Jan 24 '15 at 19:37
• It's already a closed form formula! Just replace C_BS by the Black Scholes formula – AFK Jan 25 '15 at 16:25
• It is not in the sense that you need to calculate the price twice while analytic delta just need one calculation. – mickG Jan 25 '15 at 16:28